Fourier Tickling For Homonuclear Decoupling in NMR

ABSTRACT

A method for high resolution NMR (=nuclear magnetic resonance) measurements using the application of excitation pulses and the acquisition of data points, whereby a dwell time Δt separates the acquisition of two consecutive data points, which is characterized in that one or more tickling rf (=radio frequency) pulses of duration τ p  are applied within each dwell time Δt, and that the average rf field amplitude of each of the tickling rf pulses approximately fulfills the condition  ω 1   =ω 1 τ p /Δt=πJ wherein J is the scalar J-coupling constant and ω 1 =γB 1  with γ being the gyromagnetic ratio and B 1  being the strength of the magnetic component of each tickling rf pulse. This method is effective in decoupling homonuclear couplings.

This application claims Paris Convention priority of EP 11 174 360.5 filed Jul. 18, 2011 the entire disclosure of which is hereby incorporated by reference.

BACKGROUND OF THE INVENTION

The invention concerns a method for high resolution NMR (=nuclear magnetic resonance) measurements comprising the application of excitation pulses and the acquisition of data points, whereby a dwell time Δt separates the acquisition of two consecutive data points. A method as described above is known from Andrew E. Derome, “Modern NMR Techniques for Chemistry Research”, Pergamon Press, 1987.

Double-resonance techniques were first introduced into high-resolution CW-NMR by Bloch in 1954.[1] Bloom and Shoolery[2] have shown that the application of an rf field B2 to ³¹P nuclei such that γB₂>>2π|J| can lead to the collapse of the doublet in the ¹⁹F spectrum arising from a heteronuclear coupling J(³¹P-¹⁹F). Freeman and Anderson[3, 4]proposed a theoretical description that is applicable to spin systems with either homo- or heteronuclear couplings and explains the spectral complexities and multiplicities arising from the secondary rf field B₂ while sweeping the frequency of the main rf field B₁ to observe the response in CW (=continuous wave) fashion. In particular, information about the topology of non-degenerate energy levels and the relative signs of coupling constants can be extracted.[5, 6]A detailed discussion of these effects, which have later become known as “spin tickling” experiments, has been presented elsewhere.[7, 8] Ever since, the development of advanced methods for the characterization of more and more complex systems (often in biomolecules) has been accompanied by a desire to achieve a gain in resolution and spectral simplification. The removal of homonuclear scalar interactions can simplify assignments in overlapping regions in both 1D and 2D spectra, and a number of methods have been proposed to eliminate the fine structure due to J-couplings.[9-15] However, none of these methods appear to have found widespread applications. Moreover, they usually only achieve a decoupling effect in the indirect dimension of 2D spectra. A method similar to the above is known as homonuclear decoupling, where a weak rf field of constant amplitude is applied throughout the observation of the signal. This method suffers from problems of interference between rf irradiation and signal observation, and is difficult to extend to multiple frequencies.

The object of the present invention is to present an effective and fast method of decoupling of homonuclear couplings.

SUMMARY OF THE INVENTION

This object is achieved by that one or more tickling rf (=radio frequency) pulses of duration τ_(p) are applied within each dwell time Δt, and that the average rf field amplitude of each of the tickling rf pulses is between

ω₁

=ω₁τ_(p)/Δt=^(π)/₁₀J and

ω₁

=ω₁τp/Δt=10πJ, wherein J being the scalar J-coupling constant and ω₁=γB₁ with γ being the gyromagnetic ratio and B₁ being the strength of the magnetic component of each tickling rf pulse. Decoupling of homonuclear scalar interactions in J-coupled spin systems in high-resolution NMR spectra of solutions can be achieved by applying brief but fairly intense radio-frequency (rf) “tickling” pulses in the intervals (dwell times) between the acquisition of data points. The average rf field amplitude, i.e., the peak amplitude scaled by the duty cycle, should approximately satisfy the condition

ω₁

≈πJ. It is considered sufficient if <ω₁> is between (π/10)J and 10πJ for the method to work. The method is effective over a wide range of chemical shift differences between the J-coupled pairs of nuclei.

This invention presents a 1D technique to remove homonuclear scalar interactions by applying a train of brief rf-pulses. This method may be seen as a combination of Fourier and tickling spectroscopy. In the spirit of self-deprecating acronyms such as INEPT and INADEQUATE, we like to refer to our method as Window-Acquired Spin Tickling Experiment (WASTE). The new method has been tested on proton spectra of a series of samples ranging from moderately strongly- to weakly-coupled spin systems.

A preferred variant of the inventive method is characterized in that the average rf field amplitude of each of the tickling rf pulses fulfils the condition:

ω₁

=ω₁τ_(p)/Δt=πJ.

While the method seems to work properly with values of <ω₁> only approximately reaching πJ,

ω₁

=πJ is found to be the optimal condition.

A further advantageous variant of the inventive method is characterized in that the duration τ_(p) of the tickling pulses is between 0,1 μs and 20 μs, preferably about 1 μs. Tickling rf pulses of shorter or longer duration appear to lead to unwanted side effects.

It is advantageous if the data points, which are acquired once in every dwell time, are transformed into a spectrum by a Fourier transformation.

In another preferred variant of the inventive method M tickling rf pulses of duration τ_(p) are applied at will within each of multiple dwell times Δt, each tickling rf pulse within each dwell time Δt belongs to a different comb C_(m) of tickling rf pulses, with m being a positive integer and 1≦m≦M, and all tickling rf pulses belonging to the same comb C_(m) are equidistant to each other.

Multiple tickling pulses can be applied within each dwell time and grouped into combs, within which the tickling rf pulses are equidistant to each other.

A further variation of this variant is characterized in that the phase of each tickling rf pulse belonging to the same comb C_(m) is shifted from one dwell time to the next by a constant factor. This variant allows for simultaneous decoupling of several spin systems.

The invention is shown in the drawing:

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 A scheme for Fourier tickling experiments, showing the application of a tickling rf pulse;

FIG. 2 Simulated tickling spectra stacked from bottom to top as the pulse length τ_(p) is progressively increased in steps of 2 μs until continuous spin locking is achieved;

FIG. 3 Simulated tickling spectra stacked as in FIG. 2, but with smaller increments 0.25 μs of the pulse length 0≦τ_(p)≦2.25 μs. The best decoupling occurs when τ_(p)=1 μs (fifth spectrum from the bottom);

FIG. 4 Set of simulated spectra stacked from bottom to top as the rf field strength of the tickling pulses is increased over the range 0≦ω₁/(2π)≦50 kHz in 25 steps of 2 kHz. The tickling-pulse length is τ_(p)=1 μs (duty cycle 1%), the offset Ω_(X)/(2π)=1 kHz, and the coupling constant J_(AX)=10 Hz. The true offset of spin X is indicated by the dashed line;

FIG. 5 Simulated tickling spectra stacked as in FIG. 4 but with 20 smaller steps of 0.25 kHz over a limited range 0≦ω₁/(2π)≦4.75 kHz with 1% duty cycle;

FIG. 6 Simulations for offset Ω₁/2π=0 and Ω_(S)/2π=50, 100, 150, 200, 300, 400, 500 and 600 Hz, stacked from bottom to top. The tickling pulse length was τ_(p)=1 μs. The tickling rf field strength ω₁/(2π)=2.5 kHz, with a 1% duty cycle, and the J-coupling constant was J_(AX)=10 Hz. All spectra were processed with 1 Hz line broadening;

FIG. 7 Simulated spectra similar to FIG. 6, but for a weaker rf tickling field ω₁/(2 π)=0.8 kHz;

FIG. 8 Multiplets of the off-resonance spin X calculated by average Hamiltonian theory of Eq. 2 with J_(AX)=10 Hz and Δt=100 μs. Dashed black line: unperturbed doublet in the absence of rf irradiation. Broken black line: spectrum resulting from the use of tickling pulses of duration τ_(p)=1 μs with an rf amplitude ω₁(2π)=0.5 kHz to match the condition of Eq. 6. Black continuous line: improved spectrum resulting from using twice the rf amplitude, i.e., ω₁/(2π)=1 kHz. A line broadening of 1 Hz was applied. Calculations were carried out with Mathematica;[22]

FIG. 9 Simulation of three product operators belonging to the subspace {I_(x) ^(A), 2I_(y) ^(A)I_(z) ^(X), 2I_(z) ^(A)I_(z) ^(X)} (shown as broken line with long bars, cross-line and broken line with short bars, respectively) in a two-spin system subjected to the Fourier tickling experiment of FIG. 1 with the following parameters: Ω_(A)/(2π)=0, Ω_(X)/(2π)=1 kHz, J_(AX)=10 Hz, ω₁/(2π)=3.5 kHz, τ_(p)=1 μs, Δ=100 μs, for a duty cycle of 1%, and n=2522 observation points. The norm of these three operators (closed black line) is conserved, showing that the evolution is confined to this subspace. Upon Fourier transformation, the shallow amplitude modulation of the expectation value

I_(x) ^(A)

gives rise to “tickling sidebands” as illustrated in FIG. 2. The simulations were performed with mPackages;[25]

FIG. 10 a) Conventional NMR spectrum of 2,3-dibromothiophene in DMSO-d₆ at 500 MHz. b) Tickling spectrum of the same sample with the carrier frequency set on the left-hand doublet indicated by the wavy arrow. The difference between the two chemical shifts is 305 Hz and the coupling constant J_(AX)=5.8 Hz. The tickling-pulse length was τ_(p)=1 μs for a dwell time Δt=50 μs (duty cycle 2%). The tickling field-strength was ω₁/(2π)−910 Hz. Both spectra were processed with 1 Hz line broadening.

FIG. 11 a) Conventional NMR spectrum of 2,3,6-trichlorophenol in CDCI₃. b) Tickling spectrum of the same sample with the carrier frequency set on the resonance indicated by a wavy arrow. The difference between the two chemical shifts is 98 Hz and the coupling constant J_(AX)=8.7 Hz. The tickling-pulse length was τ_(p)=1 μs for a dwell time Δt=50 μs (duty cycle 2%). The tickling field-strength was ω₁/(2π)−240 Hz. Both spectra were processed with 1 Hz line broadening. The singlet at −7.3 ppm stems from the residual CHCl₃ of the solvent;

FIG. 12 a) Conventional NMR spectrum of the A₂M₂X₃ system of propan-1-ol in D₂O at 500 MHz. b) Tickling spectrum obtained if the carrier frequency (wavy arrow) is set on the A₂ resonance. The inset shows the simplification of the multiplet of the coupling partner M₂, which has become a simple quadruplet. c) Tickling spectrum with the carrier frequency (wavy arrow) set on the X₃ resonance. The inset shows the simplification of the multiplet of the coupling partner M₂, which now appears as a simple triplet. d) Tickling spectrum with the carrier frequency (wavy arrow) set on the central multiplet of M₂, leaving three fully decoupled singlets. The tickling-pulse length was τ_(p)=1 μs and the tickling field strength was ω₁/(2π)−1.6 kHz in all cases. All spectra were processed with 1 Hz line broadening except for the expansions shown in the insets, for which no line broadening was used;

FIG. 13 a) Simulation of a conventional NMR spectrum of the AMX system at 400 MHz. b) Tickling spectrum obtained with a two-pulse tickling experiment where the carrier frequency (wavy arrow) is −40 Hz off resonance from the central resonance of spin M. A fully-decoupled spectrum is obtained by incrementing the phase of each tickling pulse according to the desired offset to be irradiated. The tickling-pulse lengths were τ_(p)=1 μs for a duty cycle 1% and the tickling field strength was ω₁/(2π)=1.5 kHz for both pulses. All spectra were processed with 1 Hz line broadening.

DESCRIPTION OF THE PREFERRED EMBODIMENT

FIG. 1 shows a scheme for Fourier tickling experiments. The tall rectangle represents the initial excitation pulse whereas the small rectangle represents a tickling pulse of duration τ_(p) applied along the x axis. The black dot represents the acquisition of a single data point. If the receiver runs continuously, this data point is obtained by averaging over all data points acquired in the interval during which the receiver is activated. The Driven Induction Decay (DID) is built up by acquiring n data points through an n-fold repetition of the loop.

Decoupling of homonuclear scalar interactions J_(AX) in coupled spin systems can be achieved by applying brief but fairly intense radio-frequency (rf) “tickling” pulses (typically with a duration τ_(p)=1 μs and an rf amplitude 2<ω₁/(2π)<3 kHz) between the acquisition of data points. These are separated by intervals (“dwell times”) that may be typically Δt=100 μs if the desired spectral width is 10 kHz. As will be explained below, the average rf field amplitude

ω₁

=ω₁τ_(p)/Δt should approximately satisfy the condition

ω₁

=πJ. If one “tickles” spin A by placing the carrier frequency of on the chemical shift Ω_(A), the fine-structure due to J_(AX) collapses in both A and X multiplets. In “linear” AMX three-spin systems with J_(AX)=0 Hz, tickling the central spin M eliminates both interactions J_(AM) and J_(MX). As a result, all three multiplets collapse and the spectrum shows only three singlets at the isotropic chemical shifts Ω_(A), Ω_(M) and •Ω_(X). However, in a general AMX three-spin system with if J_(MX)≠0 where only spin M is tickled, the multiplicity due to J_(AX) is retained for both non-irradiated A and X spins.

In the sequence shown in FIG. 1, we consider the case where the phases of the initial 90° pulse and the tickling pulses are along the y and x axes, respectively. As usual in Fourier spectroscopy, the signal is observed at regular intervals Δt (dwell times) that are inversely proportional to the desired spectral width, so as to fulfil the Nyquist condition. A tickling pulse of length τ_(p) and constant rf amplitude ω₁=γB₁ is applied near the middle of each Δt interval. A so-called Driven Induction Decay (DID) can be acquired by repeating the loop n times until one obtains the desired number of points. Numerical simulations have been performed with the SIMPSON program.[16] Relaxation effects have not been taken into account in this study. The rf carrier of the tickling pulses is set on resonance with the spin A, i.e. ω_(rf)=ω_(A), so that its offset vanishes, Ω_(A)=Ω_(A)−Ω_(rf)=0. Therefore, in the Zeeman frame rotating in synchronism with Ω_(rf)=Ω_(A), the Hamiltonian is:

H_(Tot) = Ω_(X)I_(z)^(X) + 2 π JI^(A)I^(X) + ω₁(t)(I_(x)^(A) + I_(x)^(X)),

where the third term vanishes except during the tickling pulse.

FIG. 2 shows a simulated tickling spectrum stacked from bottom to top as the pulse length τ_(p) is progressively increased in steps of 2 μs until continuous spin locking is achieved. The rf field strength is ω₁/(2π)=2.5 kHz. The offsets are Ω_(A)/(2π)=0 Hz and Ω_(X)/(2π)=1 kHz, the coupling constant is J_(AX)=10 Hz. The offset of spin X is indicated by a dashed line. All spectra were processed with 1 Hz line broadening because the on-resonance line would otherwise be very narrow and tall.

The evolution of the density matrix can be evaluated numerically using the Liouville-von Neumann equation.[17] In FIG. 2 we explored the effect of the length τ_(p) of the tickling pulses by progressively increasing the pulse length τ_(p) while keeping the dwell time Δt constant. As the length τ_(p) of the tickling pulse is increased, signals appear that are symmetrically disposed on either side of the signal of the on-resonance spin A at Ω_(A)=0 Hz. The offset of the off-resonance spin X is also perturbed, and appears to be “pushed away” from the carrier frequency. This is a manifestation of the Bloch-Siegert effect.[18-20] The apparent chemical shift is:

$\Omega_{X}^{App} = {\sqrt{\Omega_{X}^{2} + {\langle\omega_{1}\rangle}^{2}} = {\Omega_{X}{\sqrt{1 + \frac{{\langle\omega_{1}\rangle}^{2}}{\Omega_{X}^{2}}}.}}}$

Taking the first two terms of a series expansion around

^(ω) ¹

/_(Ω) _(X) =0 yields, since

ω₁

<<Ω_(X):

${{\Omega_{X}^{App} \approx {\Omega_{X}\left( {1 + {\frac{1}{2}\frac{{\langle\omega_{1}\rangle}^{2}}{\Omega_{X}^{2}}}} \right)}} = {\Omega_{X} + \frac{{\langle\omega_{1}\rangle}^{2}}{2\; \Omega_{X}}}},$

where the ratio <ω₁>²/(2Ω_(X)) gives the systematic error in rad/s. Typically, we may have an rf duty cycle τp/Δt=0.01=1% if τ_(p)=1 μs and Δt=100 μs for a spectral width of 10 kHz. If we consider an rf amplitude ω₁/(2π)=2.5 kHz and coupling partners with offsets Ω_(X)/(2π)>1 kHz (i.e., above 2 ppm at 500 MHz, or beyond 1 ppm at 1 GHz), we have systematic errors:

0<

ω₁

²/(4πΩ_(X))<0.3125 Hz.  (3)

In other words, the apparent offset of the off-resonance spin X is barely perturbed. If desired, the apparent chemical shifts observed in tickling spectra may be corrected for these Bloch-Siegert effects:

Ω_(X)≈Ω_(X) ^(App)/[1+

ω₁

²/(2Ω_(X) ²)]≈Ω_(X) ^(App)/[1+

ω₁

²/(2(Ω_(X) ^(App))²)],  (4)

where we have simply replaced Ω_(X) by Ω_(X) ^(App) on the right-hand side. The signals that appear in a symmetrical position with respect to the carrier frequency in FIG. 2 can be explained by the fact that the projection of the trajectory of the magnetization on the equatorial plane of the rotating frame is elliptical (in contrast to the circular trajectory that prevails in the absence of a tickling field) so that is can be decomposed into two counter-rotating components with unequal amplitudes. If Ω₁>>Ω_(X), the evolution of the magnetization associated with both spins would be completely suppressed in the limit of continuous irradiation when

ω₁

=ω₁ (spin-locking), so that one expects a single unmodulated signal at Ω_(A)=0 Hz. The top spectra in FIG. 2 approach this limiting case, where the two spins are in effect magnetically equivalent.

As can be seen in FIG. 3, which shows a partial enlargement of FIG. 2, an ideal decoupling effect is achieved for both resonances when the tickling pulse length is τ_(p)=1 μs. If the pulses have a decreasing duration τ_(p) each singlet appears to be flanked by two “tickling sidebands” with increasing amplitude. Note that the multiplet structures of both on- and off-resonance spins A and X remain remarkably similar for 0≦τ_(p)≦2.25 μs.

The role of the rf field strength was further investigated by simulations, as shown in FIG. 4. The tickling pulse length was kept fixed at τ_(p)=1 μs and offset Ω_(X)/(2π)=1 kHz while the field strength was progressively increased in the range 0≦ω₁/(2π)≦50 kHz. While the A spin resonance appears neatly decoupled regardless of the rf field strength, the off-resonance X spin shows a misleading splitting as the rf field strength is increased beyond 12 kHz. The distortion of the X spin resonance becomes worse as the rf field strength is increased.

However, as may be appreciated in the blown-up view of FIG. 5, the tickling sidebands are largely suppressed in the range 2<ω₁/(2π)<4 kHz if τ_(p)=1 μs and Ω_(X)/(2π)=1 kHz. The efficiency of decoupling also depends on the offset of spin X. Numerical simulations indicate that decoupling fails in strongly-coupled spin systems, i.e., if the chemical-shift difference is smaller than, say, 10 Hz.

FIG. 6 shows a set of tickling spectra for offsets 50<Ω_(X)/(2π)<600 Hz with an rf field strength ω₁/(2π)=2.5 kHz. A progressive improvement in decoupling, i.e., an increase of the intensity of the X signal is observed as the chemical shift difference is increased. The decoupling efficiency seems to be compromised when this difference is smaller than 100 Hz, i.e., smaller than 0.2 ppm at 500 MHz (11.7 T).

In FIG. 7 similar simulations are shown for a lower rf field strength ω₁/(2π)=800 Hz, again with a 1% duty cycle. Although the intensities of the tickling sidebands are much larger than in FIG. 6, adequate decoupling can be achieved for offsets as small as 50 Hz, or 0.1 ppm at 500 MHz. Thus, paradoxically, weak rf tickling strengths are required to achieve efficient decoupling in strongly-coupled spin systems.

The decoupling effect of spin tickling can be rationalized in terms of Average Hamiltonian Theory.[21] Simulations performed with Mathematica[22] show that the decoupling effect is already observed when only the zeroth-order term of the Magnus expansion that describes the pulse sequence of FIG. 1 is considered. The matrix representation of this term in the product base of a two-spin system is:

$\begin{matrix} {{\overset{\_}{H}}^{(0)} = {\begin{pmatrix} a & c & c & 0 \\ c & {- a} & {a - b} & c \\ c & {a - b} & b & c \\ 0 & c & c & {- b} \end{pmatrix}.}} & (5) \end{matrix}$

The off-diagonal elements c=ω₁τ_(p)/(2Δt)=

ω₁

/2 correspond to tickling pulses with phase x and are proportional to the duty cycle τ_(p)/Δt. The other elements a=(Ω_(X)+πJ)/2 and b=(Ω_(X)−πJ)/2 describe the offset and J-coupling interactions and do not depend on the duty cycle since the evolution under these interactions occurs both during the free precession intervals and during the tickling-pulses. The eigenvalues of this matrix represent the energy levels of the two-spin system when the expectation values of the spin operators are sampled stroboscopically with a period Δt.

The transition frequencies are given by differences of eigenvalues. In the absence of tickling pulses, the frequency difference Δv between the two single-quantum transitions associated with each spin amounts to the coupling constant J_(AX), i.e., Δv=2πJ_(AX). Identifying values of c which lead to a frequency difference Δv(c)=0 amounts to finding the condition where the J_(AX)-splitting collapses. Although the eigenvalues of H ⁽⁰⁾ are rather involved, solving the equation Δv(c)=0 yields a compact result:

$\begin{matrix} {{\langle\omega_{1}\rangle} = {{\omega_{1}\frac{\tau_{p}}{\Delta \; t}} = {{\pm \pi}\; {J_{AX}.}}}} & (6) \end{matrix}$

Thus, for a given coupling constant J_(AX), it is possible to choose an average rf field strength <ω₁>, i.e., a peak rf field strength wand a duration of the tickling pulses τ_(p) so that the splitting vanishes, provided the observation is periodic and occurs at intervals Δt. Equation 6 can be recast to give β=ω₁τ_(p)=±πJ_(AX) Δt. Thus, the flip angle β of each tickling pulse must be equal to half the angle through which the single-quantum coherences of the two components of each doublet would evolve with respect to each other in the absence of any rf perturbation under the J-coupling interaction in the dwell time Δt.

FIG. 8 shows numerical calculations of the multiplet of the off-resonance spin X using the average Hamiltonian of Eq.5 starting with an initial density operator I_(x) ^(A)+I_(x) ^(X). The tickling pulse length was τ_(p)=1 μs and the coupling constant J_(AX)=10 Hz. When ω₁/(2π)=0 Hz, an unperturbed doublet is observed (black dotted line). The multiplet obtained when the condition of Eq.6 is met with τ_(p)=1 μs, Δt=100 μs and ω₁/(2π)=0.5 kHz, is shown by a broken line: a central peak with an intensity comparable to the peaks of the unperturbed spectrum appears at Ω_(X)=1 kHz, albeit slightly displaced by a small Bloch-Siegert effect. In addition, two tickling sidebands with intensities that are about half of the singlet peak appear. When the rf amplitude is increased to Ω₁/(2π)=1.0 kHz (shown by a continuous line) the tickling sidebands move away symmetrically from Ω_(X) and lose intensity.

Although the condition Δv(c)=0 of Eq. 6 is violated by a factor of 2, the intensity of the central peak is enhanced. Thus, when the rf amplitude ω₁ is increased, the splitting between the single-quantum transitions induced by a violation of Eq. 6 does not significantly broaden the central line, while the tickling sidebands are reduced. The integrals of all three spectra are conserved.

The requirement that the scalar coupling term in the average Hamiltonian of Eq. (1) be made ineffective (on the condition that the sampling be stroboscopic) implies that the degrees of freedom of the evolution of the density operator must be severely curtailed. In simple terms, if we start with in-phase terms such as I_(x) ^(A), I_(y) ^(A), I_(x) ^(X) and I_(y) ^(X), efficient decoupling means that it should be made impossible to convert these initial states into anti-phase terms such 2I_(x) ^(A)I_(z) ^(X), 2I_(y) ^(A)I_(z) ^(X), 2I_(z) ^(A)I_(x) ^(X) and 2I_(z) ^(A)I_(y) ^(X). It turns out that, if we start with I_(x) ^(A), and if the offset Ω_(A) vanishes, coherence transfer is constrained to a Liouville subspace spanned by a triad of non-commuting operators {I_(x) ^(A), 2I_(y) ^(A)I_(z) ^(X), 2I_(z) ^(A)I_(z) ^(X)}.

FIG. 9 shows a simulation of the time dependence of these three product operators and of their norm, defined as N=(

I_(x) ^(A)

²+

2I_(y) ^(A)I_(z) ^(X)

²+

2I_(z) ^(A)I_(z) ^(X)

²)^(1/2) (continuous line) during a typical tickling experiment. Since the norm of these three operators is constant, coherence transfer must be confined to the subspace spanned by the triad of non-commuting operators. If we start with the in-phase term I_(x) ^(A), the J-coupling tends to convert it into an anti-phase operator 2I_(y) ^(A)I_(z) ^(X), but this process is stopped by the transformation of 2I_(y) ^(A)I_(z) ^(X) into longitudinal two-spin order 2I_(z) ^(A)I_(z) ^(X) due to the tickling pulses. As a result, the oscillations of the in-phase term I_(x) ^(A) are kept to a minimum. This amounts to successful decoupling. Similar phenomena were observed in a different context and dubbed “stabilization by interconversion within a triad of coherences under multiple refocusing pulses” (SITCOM).[23, 24] Of course, in the tickling experiments presented here, the brief pulses do not have any refocusing effect, but the stabilizing effect is similar.

Although the analogies may seem far-fetched, Fourier tickling achieves similar effects as repeated projective measurements, where the system is not confined to a single state, but evolves under the action of its Hamiltonian in a multidimensional subspace of Hilbert space. It may be helpful to ponder about possible variants in the light of these analogies.[26-30]

All the experiments were carried out in a static field B₀=11.7 T (500 MHz for proton). At this field, the two protons of 2,3-dibromothiophene in DMSO-d₆ are weakly coupled with (Ω_(A)−Ω_(X))/(2π)˜305 Hz and J_(AX)˜5.8 Hz.

In FIG. 10, the unperturbed spectrum is compared with the best tickling spectrum. The carrier frequency was set on the left-hand resonance, as indicated by a wavy arrow. Good decoupling and minimal interference of tickling-sidebands is obtained with a tickling field strength that was empirically optimized to ω₁/(2π)˜900 Hz.

FIG. 11 shows spectra of the strongly-coupled AB system of 2,3,6-trichlorophenol superimposed with a solvent peak marked with an asterisk.

In contrast to the AX system of FIG. 10, some tickling sidebands are clearly visible in the strongly coupled AB system of FIG. 11. This undesirable effect was highlighted in the numerical simulations of FIG. 7.

Nevertheless, a reasonable decoupling efficiency is achieved. A lower rf-field amplitude must be used for smaller differences in chemical shifts. Tickling also works if the molecule comprises magnetically-equivalent spins, as in the A₂M₂X₃ system ofpropan-1-ol (HOCH₂CH₂CH₃).

The conventional spectrum and the tickling spectra with carrier frequencies set on one of the three multiplets are shown in FIG. 12. If the carrier frequency is set on the chemical shift of A₂, the multiplicity of the coupling partner M₂ is simplified by decoupling of the ³J_(AM), though the fine structure due to the ³J_(MX) is not affected, as shown in the inset. If the carrier frequency is set on the chemical shift of X₃, the multiplicity of the coupling partner M₂ is simplified by decoupling of ³J_(XM) but the fine structure (triplet) due to ³J_(AM) remains, as shown in the inset of FIG. 12 c. Clearly tickling can decouple all J-interactions between the irradiated spin and its J-coupling partners. Obviously, tickling does not affect couplings between spins that are not irradiated. In FIG. 12 d, the carrier frequency was set on the M₂ resonance of the A₂M₂X₃ system. Since the M₂ spins are coupled to both A₂ and X₃, while ³J_(AX)=0, all three resonances appear decoupled in this case. These results show that the presence of magnetically-equivalent spins does not compromise the performance of tickling experiments. Furthermore, in all three samples considered, we did not observe any anomalies of the integrals in the tickling spectra.

It is well known that the effective frequency of the centreband of a comb C of pulses can be shifted at will from the carrier frequency v_(rf) to v=v_(rf)+φ/(2πΔt) by advancing the phase of the k^(th) pulse of the comb C in the k^(th) dwell time Δt through a shift kΔφ. Such phase modulation schemes have been used in conjunction with so-called ‘delays alternating with nutation for tailored excitation’ (DANTE).[31-34] Since the position within the interval Δt of the tickling pulses of duration τ_(p) belonging to any one comb C is immaterial, one can readily superimpose several combs C_(m) of tickling pulses with m=1, 2, . . . M, each of which may be associated with its own phase shift kΔφ_(m) and hence its own frequency shift. Each dwell time Δt does contain only one tickling pulse per comb. The tickling pulses within a dwell time Δt which belong to different combs C_(m) do not need to be equidistant. This allows one in effect to irradiate simultaneously at a manifold of frequencies, as shown in the simulations of FIG. 13.

The method is effective over a broad range of chemical shifts. Groups of magnetically equivalent spins as occur in methylene and methyl groups can be decoupled efficiently. A considerable gain in resolution and spectral simplification can thus be obtained without distortion of signal integrals. We believe that this new experimental tool can aid the characterization of complex systems, including biological macromolecules. By inserting a manifold of polychromatic tickling pulses in each Δt interval, several subsystems can be decoupled simultaneously.

The new tickling method allows one to avoid interferences and allows irradiation at multiple frequencies to decouple several interactions simultaneously.

REFERENCES

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1. A method for high resolution NMR (=nuclear magnetic resonance) measurements, the method comprising the steps of: a) applying an excitation pulse; b) acquiring a first data point; c) waiting a dwell time Δt; d) applying, during step c), one or more tickling rf (=radio frequency) pulses of duration τ_(p), each tickling rf pulse having an average rf field amplitude between

ω₁

=ω₁τ_(p)/Δt=^(π)/₁₀J and

ω₁

=ω₁τ_(p)/Δt=10πJ, wherein J is a scalar J-coupling constant and ω₁=γB₁ with γ being a gyromagnetic ratio and B₁ being a strength of a magnetic component of each tickling rf pulse; and e) acquiring a second data point immediately following completion of step c), the dwell time Δt thereby separating acquisition of the first and the second data points.
 2. The method of claim 1, wherein an average rf field amplitude of each of the tickling rf pulses fulfills the condition:

ω₁

=ω₁τ_(p)/Δt=πJ.
 3. The method of claim 1, wherein the duration τ_(p) of the tickling pulses is between 0.1 μs and 20 μs.
 4. The method of claim 3, wherein the duration τ_(p) of the tickling pulses is about 1 μs.
 5. The method of claim 1, wherein data points, acquired once in every dwell time Δt, are transformed into a spectrum by a Fourier transformation.
 6. The method of claim 1, wherein M tickling rf pulses of duration τ_(p) are applied at will within each of multiple dwell times Δt, each tickling rf pulse within each dwell time At belonging to a different comb C_(m) of tickling rf pulses, wherein m is a positive integer and 1≦m≦M, all tickling rf pulses belonging to a same comb C_(m) being equidistant from each other.
 7. The method of claim 6, wherein a phase of each tickling rf pulse belonging to the same comb C_(m) is shifted from one dwell time to a next by a constant factor. 